Regular tetrahedron formulas and how to derive them
Where the volume, surface area, height, and face area formulas of a regular tetrahedron come from, with worked numbers for edge lengths 1 and 6.
One number defines the whole solid
A regular tetrahedron is the most compact of the Platonic solids: four vertices, six edges, and four faces that are all congruent equilateral triangles. Because every edge is identical, the single edge length fixes every other measurement. That is why this calculator asks for just one value. From it you get the volume of space enclosed, the total area of the four faces, the perpendicular height from a vertex to the opposite face, and the area of a single face, each following directly from the edge.
Face area and surface area
Start with one triangular face. An equilateral triangle of side a has area equal to the square root of three divided by four, times a squared. A regular tetrahedron has four such faces, so its total surface area is four times that, which simplifies to the square root of three times a squared. For an edge of 6, one face measures 15.588457 square units and the full surface is 62.353829 square units, exactly four times larger. This is the easiest pair of formulas to remember because the face is a shape most people already know.
Height and volume
The height is the perpendicular drop from a top vertex to the center of the opposite face. Working through the geometry gives a height of a times the square root of two thirds, roughly 0.8165 times the edge. Volume then follows from the general pyramid rule, one third of the base area times the height, which collapses to the tidy expression a cubed divided by six times the square root of two. For an edge of 6 that height is 4.898979 and the volume is 25.455844 cubic units. Doubling the edge multiplies the volume by eight, since volume scales with the cube of length.
How the pieces fit together
It is worth checking the formulas against each other. The unit tetrahedron, edge length 1, gives a face area of 0.433013, a surface area of 1.732051 (which is the square root of three), a height of 0.816497, and a volume of 0.117851. Notice the surface area is exactly four times the face area, a quick sanity check you can apply to any edge. These relationships mean you can scale a known result rather than recompute: areas grow with the square of the edge and volume with its cube.